mathematical aspects of thermoacoustics.pdf

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Mathematical Aspects ofThermoacoustics


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Copyright c2009 by P.H.M.W. in t panhuis, Eindhoven, The Netherlands.All rights are reserved. No part of this publication may be reproduced, stored in a re-trieval system, or transmitted, in any form or by any means, electronic, mechanical,photocopying, recording or otherwise, without prior permission of the author.

Printed by Print Service Technische Universiteit Eindhoven

Cover design by Jorrit van Rijt

A catalogue record is available from the Eindhoven University of Technology Library

ISBN 978-90-386-1862-3NUR 919Subject headings: thermoacoustics; acoustics; acoustic streaming; thermodynamics; per-turbation methods; numerical methods; boundary value problems; nonlinear analysis;shock waves.

This research was financially supported by the Technology Foundation (STW), grantnumber ETTF.6668.


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Mathematical Aspects ofThermoacoustics

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het Collegevoor Promoties in het openbaar te verdedigen

op donderdag 25 juni 2009 om 16.00 uur

door

Petrus Hendrikus Maria Wilhelmus in t panhuis

geboren te Roermond


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Dit proefschrift is goedgekeurd door de promotor:

prof.dr. J.J.M. Slot

Copromotoren:dr. S.W. Rienstraenprof.dr. J. Molenaar


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To Ik Cuin


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Preface

This project is part of a twin PhD program between the Departments of Mathemat-ics and Computer Science and Applied Physics and is sponsored by the TechnologyFoundation (STW), Royal Dutch Shell, the Energy Research Centre of the Netherlands(ECN), and Aster Thermoacoustics. I would like to express thanks to all people whoparticipated in this project. First and foremost I would like to thank my mathematicalsupervisors dr. Sjoerd Rienstra, prof.dr. Han Slot, and prof.dr. Jaap Molenaar for theirexpert guidance and stimulating support. I am also indebted to my physics colleaguesPaul Aben, dr. Jos Zeegers, and prof.dr. Fons de Waele, who helped to broaden anddeepen my understanding of the physics involved. I am also grateful for the manyuseful discussions I have had with the people from ECN, Aster Thermoacoustics, andShell.

My defense committee is formed by prof.dr. Anthony Atchley, prof.dr. Bendiks-JanBoersma, and prof.dr. Mico Hirschberg, together with my supervisors dr. Sjoerd Rien-

stra, prof.dr. Han Slot, and prof.dr. Jaap Molenaar. I would like to thank them for thetime invested and their willingness to judge my work. I also want to thank prof.dr. BobMattheij for agreeing to be part of the extended defense committee.

What made these four years especially enjoyable was the great working atmospherewithin CASA and the Low Temperature group. My special thanks go out to my of-fice mate and partner-in-crime Erwin who started and finished his PhD (and Master)at the same time as I did. Many thanks also to all the current and former colleaguesthat I have had the pleasure to work with, in particular the PhD students and postdocs:Aga, Ali, Andriy, Bart, Berkan, Christina, Darcy, Davit, Dragan, Hans, Jurgen, Kakuba,Kamyar, Kundan, Laura, Marco, Maria (2), Mark, Matthias, Maxim, Michiel, Miguel,Mirela, Nico, Oleg, Patricio, Paul, Remko, Remo, Roxana, Shruti, Sven, Tasnim, Temes-gen, Valeriu, Venkat, Wenqing, Yabin, Yan, Yixin, Yves, Zoran. I fondly think back to

our daily lunches at the Kennispoort, the weekly poker games, the road-trips to Den-mark, the regular squash/tennis/football games, and the many nights in town that Ihave enjoyed with so many of you. Our two secretaries Marese and Enna also deservea special word of thanks, for making life of a PhD student so much easier by taking careof all administrative details. I am also thankful to the members of the football teamsPusphaira and Old Soccers, and hope they will have more success without me.

On a more personal level, I would like to thank all my friends and family, for theircontinuous love and support. I especially want to show appreciation to my mom forher unbridled enthusiasm and my dad, who I wish could have been here today. I alsowant to thank my siblings Jos, Hellen, and Dorris and their significant others Marjanne,Joram, and Tonnie. Of course I should not forget to mention my little nephew Sep, who


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ii Preface

is getting so big now. Last, but definitely not least, I would like to thank my girlfriendJessey, to whom this thesis is dedicated, for her unlimited love and patience in these lastfew months.

Peter in t panhuisEindhoven, May 2009


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Nomenclature

General symbols and variables

A [m2] cross-sectional area

A [m2] cross-sectionb [ms2] specific body force fieldc [m s1] speed of soundCp [J kg

1 K1] isobaric specific heatCs [J kg

1 K1] specific heat of stack materialCv [J kg

1 K1] isochoric specific heatd [m] diameterf [Hz] frequency

f viscous Rott functionfk thermal Rott functionfs solid Rott functionF viscous Arnott functionFk thermal Arnott functionFs solid Arnott functionF Fourier transformG Greens function

g [ms2] gravitational accelerationH [W] total power

h [J kg1] specific enthalpyIm imaginary part

k [m1

] wave numberK [W K1 m1] thermal conductivity [m] displacementL [m] typical length

m [kg m2 s1] time-averaged mass fluxM [kg m2 s1] time-averaged volumetric mass flux

n [mol s1] molar flow rateP [W] powerp [Pa] pressurepA [Pa] pressure oscillation amplitudepamb [Pa] ambient pressure


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iv Nomenclature

Q [W] heat flow per unit time

q [W m2] heat fluxr [m] radial coordinateRe real part

Rspec [J kg1 K1] specific gas constant

R [m] radiuss [J kg1 K1] specific entropyS [J kg1] entropyS surfaceSu [K] Sutherlands constantt [s] time

T [K] temperatureU [m s1] typical fluid speedU [W] internal energy

v = (u, v, w) [m s1] velocity vectorV [m3] volumeW [W] acoustic powerx = (x, y, z) [m] spatial coordinate

Z [N s m3] impedance

[K1] isobaric volumetric expansion coefficient boundary, interface [m] penetration depth

[J kg1] specific internal energy [m] wave length [Pa s] dynamic (shear) viscosity [Pa s] second viscosity

[kg m3] density [N m2] viscous stress tensor [rad] angular coordinate

[rad s1] angular frequency of the acoustic oscillations

Dimensionless numbers

A amplitudeBr blockage ratioCOP coefficient of performanceCOPR relative coefficient of performanceCOPC Carnot coefficient of performanceDr drive ratioFr Froude numberMa acoustic Mach numberNL Lautrec number gasNs Lautrec number solidPr Prandtl numberR reflection coefficient


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Nomenclature v

Sk Strouhal number based on kWo Womersley numberW second Womersley number coefficient of nonlinearity ratio specific heats coefficient of stack dissipation deviation from resonance coefficient of weak nonlinearity efficiencyR relative efficiencyC Carnot efficiency

aspect ratio driver Mach numbers stack heat capacity ratio Helmholtz number ratio thermal conductivities porosity dimensionless viscosity

Sub- and superscripts and special operators

a dimensionlessa time averaging

a transverse averaginga per unit timea complex conjugateare f reference valuea+ top platea bottom platea0 steady zeroth ordera1 first harmonica2,0 steady second order (streaming)a2,2 second harmonicaC coldaH hot

ag gas, fluidak thermalaL thermalap isobaricaR rightas solid, source, stack centerat outera transverse componentsav isochorica viscous


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vi Nomenclature


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Contents

Preface i

Nomenclature iii

1 Introduction 11.1 A historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The basic mechanism of thermoacoustics . . . . . . . . . . . . . . . . . . . 61.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Thermodynamics 172.1 Laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Thermodynamic performance . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Refrigerator or heat pump . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Prime mover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 The thermodynamic cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Standing-wave phasing . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Traveling-wave phasing . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Bucket-brigade effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Modeling 273.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Small-amplitude and long-pore approximation . . . . . . . . . . . . . . . . 34

4 Thermoacoustics in two-dimensional pores with variable cross-section 374.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Mean temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Integration of the generalized Swift equations . . . . . . . . . . . . . . . . 51

4.4.1 Exact solution at constant temperature . . . . . . . . . . . . . . . . 524.4.2 Short-stack approximation . . . . . . . . . . . . . . . . . . . . . . . 544.4.3 Approximate solution in short wide channels . . . . . . . . . . . . 56


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viii Contents

4.5 Acoustic streaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.6 Second harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.7 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.7.1 Acoustic power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.7.2 Total power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Thermoacoustics in three-dimensional pores with variable cross-section 735.1 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Mean temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.3 Integration of the generalized Swift equations . . . . . . . . . . . . . . . . 83

5.3.1 Ideal stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.2 Rotationally symmetric pores . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Acoustic streaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.5 Second harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.6 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Standing-wave devices 936.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.3 A thermoacoustic couple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.1 Acoustically generated temperature differences . . . . . . . . . . . 976.3.2 Acoustic power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4 A standing-wave refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.5 A standing-wave prime mover . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.6 Streaming effects in a thermoacoustic stack . . . . . . . . . . . . . . . . . . 112

7 Traveling-wave devices 1157.1 A traveling-wave prime mover . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2.1 Regenerator and thermal buffer tube . . . . . . . . . . . . . . . . . . 1197.2.2 Optimization procedure . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3.2 Regenerator efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3.3 Geometry optimization . . . . . . . . . . . . . . . . . . . . . . . . . 129

8 Nonlinear standing waves 133

8.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.1.1 Kuznetsovs equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.1.2 Bernoullis equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.1.3 Perturbation expansion . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.2 Solution away from resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.2.1 Arbitrary excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408.2.2 Harmonic excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.3 Solution near resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.3.1 Exact solution when = = 0 using Mathieu functions . . . . . . 1 4 38.3.2 Steady-state solution for = O(1) . . . . . . . . . . . . . . . . . . . 1448.3.3 Steady-state solution for = O(1) . . . . . . . . . . . . . . . . . 148


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Contents ix

8.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.4.1 Nonlinear standing waves in a closed tube . . . . . . . . . . . . . . 1558.4.2 Nonlinear standing waves in a thermoacoustic refrigerator . . . . . 1 5 7

9 Conclusions and discussion 161

Appendices 165

A Thermodynamic constants and relations 165

B Derivations 167B.1 Total-energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

B.2 Temperature equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

C Greens functions 169C.1 Fj-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169C.2 Fj,2-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171C.3 Greens functions for various stack geometries . . . . . . . . . . . . . . . . 172

Bibliography 175

Index 185

Summary 187

Samenvatting 189

Curriculum Vitae 191


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x Contents


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Chapter 1

Introduction

As the name indicates, thermoacoustics combines the fields of thermodynamics andacoustics and describes the interaction between heat and sound. The term was firstcoined in 1980 by Rott [119] who described its meaning as rather self-explanatory. Ac-cording to Rott the most general interpretation of thermoacoustics includes all effectsin acoustics in which heat conduction and entropy variations of the (gaseous) mediumplay a role.

Ordinarily a sound wave in a fluid is seen as the combined effect of pressure andvelocity (displacement) oscillations, but as a response to these (isentropic) pressure os-cillations, temperature oscillations may occur as well. In free space the temperature

variations will be small and the gas parcels will expand and compress adiabatically.However, when the fluid is allowed to interact thermally with solid boundaries, heattransfer between the gas and the solid will take place and a wide range of thermoacous-tic effects may occur.

It has been realized that under the right operating conditions these thermoacousticconcepts can be harnessed and exploited to create two kinds of thermoacoustic devices:the refrigerator or heat pump that turn sound into useful refrigeration or heating, and theprime mover that turns heat into useful (acoustic) work. Typically, such devices are con-structed from straight or looped tubes with a porous medium suitably located inside.

Thermoacoustic devices can be of much practical use, because significant amountof heat and mechanical power can be produced at a reasonable efficiency. Moreover,in contrast to more conventional engines and refrigerators, they can operate without

cranks and pistons, and usually have no more than one (mechanically) moving part.Because of this and their inherent simplicity, they are very reliable, require little mainte-nance, and can be produced at relatively low cost. Furthermore, thermoacoustic devicesare friendly to the environment, as they use environmentally friendly gases, produceno toxic waste, and are easily adaptable to use solar power [27] or industrial wasteheat [127] as a driving source. Currently, research is also being done on the possibil-ities of using biomass to drive a thermoacoustic stove [112], to be used in developingcountries.

Despite all these advantages there are still some challenges left that need to be re-solved before thermoacoustic devices can be used competitively on a large scale. Firstly,due to the oscillatory nature of the flow and the interaction with solid boundaries all


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2 1.1 A historical perspective

kinds of complicated flow patterns may arise, such as vortex shedding, turbulence, orasymmetric flow. Furthermore, the heat transfer is far from ideal and entropy is created.Additionally, thermoacoustic devices often operate at high pressure amplitudes, whichcan give rise to various nonlinear effects, such as the build-up of shock waves. The com-bined effect of all these phenomena can and will degrade the performance and holdsback the development of highly efficient devices. Moreover, as long as these effects arenot understood and modeled systematically, it will be hard to make accurate theoreticalpredictions. The mathematical aspects of oscillatory gas flow with heat transfer to solidboundaries in wide or narrow pores will be the topic of this thesis.

1.1 A historical perspectiveThermoacoustics has a long history that dates back more than two centuries. The inter-est in thermoacoustics was first piqued in 1816, when Laplace showed that even Newtonwas not infallible. Laplace [72] pointed out that Newtons approximation of the speedof sound [92] was incorrect because he assumed isothermal compression and expansionof the air and did not compensate for variations in temperature. Correcting for theseeffects, Laplace found a value that was 18% higher.

Hydrogensupply

Flame

(a) Higgins singing flame

Wirescreen

Generation

of sound

Convectionflow

(b) Rijke tube

Figure 1.1: (a) Higgins singing flame: for suitable positions of the flame the tube will start toproduce sound. (b) The Rijke tube: the loudest sound is produced when the heated wire screen ispositioned at one-fourth from the bottom of the pipe.

The first records of heat-driven oscillations are the observations of Higgins [54, 102]in 1777, who experimented with an open glass tube in which acoustic oscillations wereexcited by suitable placement of a hydrogen flame, the so-called singing flame. A


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Introduction 3

similar, but more famous experiment was performed by Rijke [111] who in his efforts todesign a new musical instrument from an organ pipe, constructed the so-called Rijketube. As depicted in figure 1.1, he replaced Higgins hydrogen flame by a heated wirescreen and found that when the screen was positioned in the lower half of the opentube spontaneous oscillations would occur, which were strongest when the screen waslocated at one fourth of the pipe. The oscillations would stop if the top of the tube wasclosed, indicating that the convective air current through the pipe was necessary forsound to be produced. Higgins and Rijkes work later led to the birth of combustionscience, with applications in rocket science and weapon industry. For a full reviewon devices related to the Rijke tube we refer to Feldman [41] or more recently Raunet al. [108].

The earliest predecessor of the type of thermoacoustic prime movers considered inthis thesis is the Sondhauss tube, depicted in figure 1.2. It was invented in 1850 by Sond-hauss [126] based on an effect often noticed by glass blowers: generation of loud soundwhen a hot bulb is blown at the end of a cold narrow tube. Sondhauss found that if asteady gas flame was supplied to the closed bulb, the tube would produce a clear soundwhich was characterized by the length of the tube; the larger the bulb or the longer thetube, the lower the frequency of the sound. Unfortunately, Sondhauss did not manageto explain why the oscillations arose. Feldman [42] also reviewed most of the phenom-ena related to the Sondhauss tube as he did for the Rijke tube. An important differencebetween these two devices is that the Sondhauss tube does not require a convective aircurrent for oscillations to occur.

Bulb Tube stem

SoundFlame

Figure 1.2: The Sondhauss tube: sound is generated from the tip of the stem, when heat issupplied to the bulb.

Another early example of a thermoacoustic prime mover is the phenomenon knownas Taconis oscillations, often observed in cryogenic storage vessels. Taconis [139]cooled a gas-filled tube from room temperature to a cryogenic temperature by insertingit into liquid helium and observed spontaneous oscillations of the gas. The conditionsfor these type of oscillations have been investigated experimentally by Yazaki et al. [156].

The first qualitative explanation for heat-driven oscillations was given in 1887 byLord Rayleigh. In his classical work The Theory of Sound [109], he explains the pro-duction of thermoacoustic oscillations as an interplay between heat fluxes and densityvariations:


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4 1.1 A historical perspective

If heat be given to the air at the moment of greatest condensation (compression)or taken from it at the moment of greatest rarefaction (expansion), the vibration isencouraged.

Rayleighs qualitative understanding turned out to be correct, but a quantitatively ac-curate theoretical description of these phenomena was not achieved until much later.

The reverse process, generating temperature differences using acoustic oscillations,is a relatively new phenomenon. In 1964 Gifford and Longsworth [48] invented thepulse-tube refrigerator, by which they managed to cool down to a temperature of 150K. In their device, depicted in figure 1.3, heat was pumped along the tube wall by sup-plying pressure pulses at low frequencies. Initially it was considered nothing more thanan academic curiosity as it was highly inefficient, but current-day pulse-tube cryocool-

ers can reach efficiencies up to 20% of the ideal efficiency and temperatures as low as2 K. In fact, nowadays pulse-tube refrigeration is one of the most favored technologiesfor cryocooling . For a complete history and review of pulse-tube crycooling we re-fer to Radebaugh [104, 105]. Detailed modeling and numerical analysis of pulse-tuberefrigerators can be found in [7981].

Rotary

valve

VentHigh-pressure

source

Regenerator

Cold end

Pulsetube

Roomtemperature

Figure 1.3: The pulse-tube refrigerator of Gifford and Longsworth. The temperature is cooledfrom room temperature at the hot end to 150 K at the cold end.

Sound-driven cooling was also observed by Merkli and Thomann [88] when theyperformed experiments on cooling in a simple gas-filled piston-driven resonator. To ex-plain these effects an extended acoustic theory was developed which predicted coolingin the tube where the velocity amplitude was at its maximum, in good agreement withthe experiments.

A crucial advance in experimental thermoacoustics came in 1962 when Carter et al.[22] realized that the performance of the Sondhauss tube could be improved by suitableplacement of a porous medium inside, in the form of a stack of parallel plates. Thepresence of a stack, with heat transfer from one end to the other, makes it much easierto produce a significant temperature difference and will be the essential ingredient forthe kind of thermoacoustic devices considered in this thesis.

The foundation for theoretical thermoacoustics was laid in 1868 by Kirchhoff [66],


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Introduction 5

who investigated the acoustic attenuation in a duct due to oscillatory heat transfer be-tween the isothermal tube wall and the gas inside the tube. His results were generalizedby Kramers [67] for a tube supporting a temperature gradient. The breakthrough camein 1969 when Rott et al. , inspired by the Taconis oscillations, started an impressive se-ries of articles [91, 115118, 120, 121], in which a linear theory of thermoacoustics wasderived. Rott abandoned the boundary-layer approximation as used by Kirchhoff andKramers, and formulated the mathematical framework for small-amplitude dampedand excited oscillations in wide and narrow tubes with an axial temperature gradient,only assuming that the tube radius was much smaller than the length of the tube. In1980 Rott summarized his results in a review work [119], which became the cornerstonefor the subsequent intensified interest in thermoacoustics.

In the eighties a very intensive and successful research program was started at theLos Alamos National Laboratory by Wheatley, Swift, and coworkers [133,151, 152]. Us-ing Rotts theory of thermoacoustic phenomena they started to design and build practi-cal thermoacoustic devices. Important was Hoflers invention of a standing-wave ther-moacoustic refrigerator [55,56], which proved that Rotts theoretical analysis was cor-rect. Hoflers refrigerator, shown in figure 1.4, used a loudspeaker to drive a closedresonator tube with a stack of plates positioned near the speaker. At the other end ofthe tube a resonator sphere was attached to simulate an open ending, so that effectivelyone can speak of a quarter-wave-length resonator. Inside the refrigerator a standing-wave is maintained by the speaker, generating a temperature difference across the stacksuch that heat is absorbed at the low temperature or waste heat is released at the hightemperature.

Hot heat exchanger

Cold heat exchanger

Driver Stack

Resonatorsphere

Figure 1.4: Hoflers standing-wave refrigerator. The hot end of the stack is thermally anchoredat room temperature and the standing wave generates cooling at the cold end of the stack.

A whole new branch of thermoacoustic devices started in 1979 with Ceperleys real-ization [23, 24] that thermoacoustic devices based on the Stirling cycle [21] with idealheat transfer, could reach much higher efficiencies than devices based on standing-wave modes of operation. His idea was to design machines that allow a traveling waveto pass through a dense porous medium (the regenerator) using a toroidal geometry.Yazaki et al. [155] managed to build a traveling-wave prime mover based on these prin-ciples, but at very low efficiency due to large viscous losses. Finally, Backhaus andSwift [14] managed to overcome these problems by designing a traveling-wave primemover (shown in figure 1.5) that combines the toroidal geometry with a resonator tubeto reduce the velocities in the loop.

Swift was the first to give a comprehensive analysis of thermoacoustic devices in his


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6 1.2 The basic mechanism of thermoacoustics

Resonator

Load

Jet pump

Regenerator

Figure 1.5: Schematic drawing of the traveling-wave prime mover of Backhaus and Swift. Thesound produced by the regenerator is absorbed by an acoustic load that is attached to the regen-erator.

review article [131] based on Rotts work. He also gives a detailed description of thethermodynamics involved, a complete historical overview, experimental results, and hetreats several types of devices. Since then Swift and others have contributed much to thefurther development and analysis of thermoacoustic devices. Most of the literature hasbeen collected and summarized in Garretts review work [45]; in particular we mentionSwifts textbook [135], which provides a clear introduction into thermoacoustics. Lastly,we note that several articles have been written as well [15, 47, 133, 150, 153], aimed atreaders new to the field of thermoacoustics, while various educational animations canbe found at the website of Los Alamos National Laboratory [130].

1.2 The basic mechanism of thermoacoustics

The thermoacoustic principles can be understood best by following a given parcel offluid as it oscillates near a solid boundary. We start by considering fluid parcels oscillat-ing far away from the wall in a closed tube supporting a standing wave or in a infinitetube supporting a traveling wave, as depicted in figure 1.6. Note that under isentropicconditions the pressure oscillations are accompanied by temperature oscillations, whichis used in figures 1.6(c) and 1.6(d) to draw schematically the temperature-position cyclethat a fluid parcel undergoes during one oscillation.

Consider a fluid parcel oscillating in the closed tube, far away from the tube wall. Inthe center of the tube we have a pressure node and a velocity antinode. As a result theparcel will undergo large displacement without temperature variations. On the otherhand, near the ends of the tube we have a pressureantinode and a velocity node, and theparcel will almost stand still and undergo large temperature variations, giving a verysteep temperature gradient. If the parcel is oscillating away from a velocity or pressurenode, then a finite nonzero temperature gradient will arise. Within thermoacoustics theslope of these adiabatic temperature variations is known as the critical temperaturegradient.

In the infinite tube the situation is somewhat different, since the pressure and ve-locity oscillations are in phase for all positions in the tube. As a result the parcels al-ternately move to the right with a high temperature and then to the left with a lowertemperature, giving circle-like temperature-position cycles. It follows that when a smalltraveling-wave component is added to a standing wave, or vice versa, the thermody-


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Introduction 7

(a) (b)

T

x

(c)

T

x

(d)

T

x

(e)

Figure 1.6: Gas parcels oscillating (a) with standing-wave phasing in a closed tube and (b) withtraveling-wave phasing in an infinite tube. The pressure oscillations are indicated by dashedlines, the velocity oscillations by solid lines, and the arrows give the direction of oscillation. (c)The temperature of the gas parcel during one oscillation as a function of its position relative to the

standing wave. (d) Temperature-position diagram for traveling wave. (e) Temperature-positiondiagram when a small traveling-wave component is added to the standing wave.

namic cycles will turn into tilted ellipses as shown in figure 1.6(e).Suppose we look at the same parcels, but now oscillating near a wall supporting a

temperature gradient in axial direction. We consider two cases:

(a) The tube wall supports a much smaller temperature gradient than the critical tem-perature gradient of the fluid parcel. As a result heat will be transported from thegas parcel into the wall at the hot end and from the wall into the gas parcel at thecold end (figure 1.7(a)). The transport of heat from the low temperature to the hottemperature will require the input of acoustic work. This is the condition for a

refrigerator or heat pump.

(b) The tube wall supports a much larger temperature gradient than the critical tem-perature gradient of the fluid parcel. As a result heat will be transported from thewall into the gas parcel at the hot end and from the gas parcel into the wall at thecold end (figure 1.7(b)). The transport of heat from the high temperature to thelow temperature will produce acoustic work as output. This is the condition for aprime mover.

This is in a nutshell the basic mechanism of thermoacoustics. In the next chapter wewill analyze the combined effect of oscillations and heat transfer in more detail usingBrayton [21] and Stirling cycles [145]. Specifically we will show how the distance to


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8 1.3 Classification

the solid and the phasing of pressure and velocity affect the heat transfer between gasand solid.

T

position parcel

Q

Q

parcelwall

(a) Refrigerator

T

position parcel

Q

Q

parcelwall

(b) Prime mover

Figure 1.7: The temperature-position diagram for the adiabatic parcel-temperature oscillations() and the wall temperature (- -). In (a) we consider a refrigerator and apply a small tempera-ture difference and in (b) we consider a prime mover with large temperature difference across thewall. The arrows show the transport of heat Q from the wall to the gas parcel and vice versa.

1.3 Classification

We consider thermoacoustic devices of the type shown in figure 1.8, that is, a possiblylooped duct containing a fluid (usually a gas) and a porous solid medium, if necessarywith neighboring heat exchangers. In addition loudspeakers or other sources of soundmay be attached to the ends of the tubes. The porous medium is modeled as a collectionof narrow arbitrarily shaped pores aligned in the direction of sound propagation. Typ-ical examples are parallel plates and circular or rectangular pores. As will be discussedbelow, thermoacoustic devices can be divided [45] into several categories: heat-drivenversus sound-driven devices, standing-wave versus traveling-wave devices, or stack-based versus regenerator-based devices.

Prime mover vs. refrigerator

We distinguish heat-driven and sound-driven devices. A thermoacoustic prime moverabsorbs heat at a high temperature and exhausts heat at a lower temperature whileproducing acoustic work as an output. A refrigerator or heat pump absorbs heat at a lowtemperature and requires the input of acoustic work to exhaust more heat to a highertemperature. The only difference between a heat pump and a refrigerator is whetherthe purpose of the device is to extract heat at the lower temperature (refrigeration) or toreject heat at the higher temperature (heating). Therefore, in this thesis we will use theterm refrigerator loosely to refer to either of these devices.


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Introduction 9

(a) straight duct (b) looped duct

Figure 1.8: Schematic view of two possible duct configurations: straight or looped.

In the literature the term thermoacoustic engine is used as well, either to indicate athermoacoustic prime mover or as a general term to describe all thermoacoustic devices.To avoid confusion we will refrain from using this term.

Stack-based devices vs. regenerator-based devices

A second classification depends on whether the porous medium used to exchange heat

with the working fluid is a stack or a regenerator. Typically inside a regeneratorthe pore size is much smaller than inside a stack. Garrett [45] uses the so-called Lautrecnumber NL to indicate the difference between a stack and regenerator. The Lautrecnumber is defined as the ratio between the half pore size and the thermal penetration

depth1. Gas parcels that are separated from the wall by a distance much larger than thethermal penetration depth, will have no thermal contact with the wall. If NL 1 theporous medium is called a stack and the gas parcels in the stack will have imperfectthermal contact with the solid. If NL 1, then the porous medium is called a regen-erator, and the gas parcels inside the regenerator will have perfect thermal contact withthe solid. Whenever the pore size is unknown or irrelevant, the porous medium will bereferred to as stack.

Standing-wave devices vs. traveling-wave devices

Lastly, thermoacoustic devices can also be categorized depending on the phase shiftbetween the pressure and velocity oscillations at the location of the stack. In a closedand empty resonator a pure standing-wave can be maintained and the pressure andvelocity oscillations will be exactly 90 degrees out of phase. In an empty infinite tube(or a loop) a pure traveling-wave can be maintained, so that the pressure and velocityoscillations are exactly in phase. As soon as we insert a stack in either of these tubes,the phasing between pressure and velocity will change because of partial reflection atthe stack interfaces. Moreover, if we consider a looped tube of the type depicted in

1The thermal penetration depth is the distance heat can diffuse through within a characteristic time


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10 1.4 Applications

figure 1.8(b), with a resonator tube attached to it, the phasing will be affected evenmore. However, in practise both the straight geometry depicted in figure 1.8(a) and thelooped geometry depicted in figure 1.8(b) can be chosen such that the sound field at thestack is predominantly a standing wave or a traveling wave.

In Section 2.3 we will show that it is beneficial to use a stack inside a standing-wavedevice and a regenerator inside a traveling-wave device. For this reason thermoacousticdevices are usually classified as either standing-wave stack-based devices or traveling-wave regenerator-based devices.

1.4 Applications

Over the years thermoacoustic devices have found applications in areas like food in-dustry, defense industry, spacecraft, telecommunication, electronics, energy sectors, andconsumer products. Some of these devices are heat-driven, some are sound-driven, andothers combine the two effects. In this section we will treat a few examples that havebeen or will be used commercially. Most of these applications are motivated by thequest for reliable, cheap, or environmentally friendly sources of energy.

Down-well power generation

The natural-gas industry use sensors to measure properties of the gas that streamsthrough subterranean gas pipes. These sensors are located far below ground and needelectrical power, which has to be delivered via batteries or long cables. The reliability

of such equipment is usually quite poor, requiring costly repairs or replacements due todifficult accessibility and extreme operating conditions. Since most wells are used formany years, there is a necessity for a cheap and reliable alternative for power genera-tion, which thermoacoustics can provide.

stack

main flow

Figure 1.9: Schematic drawing of the side-branch system. A standing wave is generated in theside branches due to the interaction of the main flow, supplied by the pipeline, with the edges ofthe side branch. If a stack is suitably positioned in the standing wave, a temperature differencecan be generated.

The answer lies in a technique suggested by Slaton and Zeegers [123125], whichavoids the use of moving parts and uses part of the main-flow energy in the gas pipes


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Introduction 11

to generate aero-acoustic sound. In their experimental set-up, shown in figure 1.9, theyattach two side branches to the main pipeline. Due to the interaction of the main flowwith the edges of the side branch, vortices will be created. By adjusting the lengthsof the side branch to the flow speed, it is possible to match the frequency of the vor-tex shedding with the fundamental standing-wave mode of the side branch, and astanding-wave sound field is created in the side branch. Finally, by suitable placementof a stack a temperature difference can be generated, which can be used to produceelectrical power with thermoelectric elements. The flow patterns and vortex sheddingin such side-branch systems have been visualized experimentally and numerically byKriesels et al. [68] and Dequand et al. [36].

Upgrading of industrial waste heat

Another important potential application for thermoacoustics is the upgrading of indus-trial waste heat. Huge amounts of heat produced by industry remain unused becauseof small power outputs or temperatures that are too small. The Energy research Centreof the Netherlands (ECN) has developed a thermoacoustic system that uses part of thewaste heat to power a prime mover that drives a heat pump to upgrade the temperatureof the remainder. The apparatus is shown in figure 1.10.

Prime MoverHeat Pump

Resonator

Figure 1.10: Upgrading of industrial waste heat using a combination of a prime mover and heatpump. Part of the waste heat is used to power the prime mover, which generates acoustic powerto drive the heat pump and heat the remaining part of the waste heat.

Thermoacoustic cryocooling

Thermoacoustics is the only technology that can cool to temperatures close to the abso-lute zero without using moving parts and is therefore very interesting for applicationsrequiring crycooling. One such application is the liquefaction of natural gases whichrequires very low temperatures. At the Los Alamos National Laboratory (LANL) aheat-driven thermoacoustic refrigeration system [134, 138] has been designed capableof liquefying natural gases. Their system, depicted in figure 1.11, uses a toroidal geom-etry attached to a long resonator tube, with a prime mover located in the toroidal partand a refrigerator located near the end of the resonator. Part of the natural gas is burnedto supply heat which is converted into acoustic power by the prime mover. The acousticpower is then provided to the refrigerator and subsequently used to cool the remainderof the natural gas until it is liquefied.

Thermoacoustic cyrocooling is also applied to the cooling of electronics. The Amer-ican Navy used a Shipboard Electronics ThermoAcoustic Chiller (SETAC) to cool elec-tronics on board of one of their destroyers [86]. A similar application found its way into


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12 1.5 Thesis overview

PrimeMover

Refrigerator

Figure 1.11: Schematic drawing of a heat-driven thermoacoustic refrigeration system. By burn-ing part of the natural gases at the prime mover, the remainder can be cooled to liquefaction atthe refrigerator.

spacecraft, when in 1992 the Space ThermoAcoustic Refrigerator (STAR) was launchedon the Space Shuttle Discovery. It was the answer [46] to the need for reliable, com-pact, and long-lived spacecraft cryocooling for the cooling of sensors aboard the shut-tle. Other applications of thermoacoustics within spacecraft concern the developmentof thermoacoustic systems suitable for electricity generation on space missions [16,140].

Food refrigeration and airconditioning

Thermoacoustics refrigerators can also be used to replace conventional food refriger-ators or airconditioning, without the harmful polluting side-effects. One well-knownexample is the collaboration [101] between Pennsylvania State University and Ben andJerrys Homemade Ice Cream, that led to the development of a thermoacoustic in-storeice-cream cabinet, capable of keeping ice cream at -18 C. The aim was to find a cost-competitive and reliable alternative to the use of polluting refrigerants and thereby re-duce the emission of global-warming gases.

Cooking stove

Recently the SCORE (Stove for Cooking, Refrigeration and Electricity) project has started,an international research program led by the University of Nottingham [112]. The aimis to develop a biomass-powered thermoacoustic system, to be used as an affordable,safe, and efficient alternative for the energy needs of third-world countries. Currentsources of energy, like open fires, are highly inefficient and can produce serious health

hazards. The SCORE stove will serve as a versatile domestic appliance, being a cooker,a refrigerator, and an electricity regenerator all in one.

1.5 Thesis overview

In this thesis we will derive a weakly nonlinear theory of thermoacoustics, applicableto wide and narrow pores with arbitrarily shaped cross-sections that may vary slowlyin axial direction. We will use dimensional analysis and small-parameter asymptoticsso that (quadratically) nonlinear terms can be systematically included. The use of adimensionless framework has two main advantages:


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Introduction 13

Nondimensionalization allows us to analyze limiting situations in which param-eters differ in orders of magnitude, so that we can study the system as a functionof dimensionless parameters connected to geometry, heat transport, and viscouseffects.

We can give explicit conditions under which the theory is valid. Furthermore, wecan clarify under which conditions additional assumptions or approximations arejustified.

In the end we will apply this theory to model, analyze, and simulate both standing-wave and traveling-wave devices and we will show how this approach can be used asan aid for optimizing their design.

The main ingredients of this thesis are therefore the choice of geometry for the stackpores, the use of dimensionless parameters, the inclusion of quadratic nonlinearities,and the modeling and numerical simulation of complete devices. In Section 1.6 we willdiscuss briefly the relevant literature on each of these topics, but first we will give achapter-by-chapter overview of the contents of this thesis.

In Chapter 2 the thermodynamics of thermoacoustics will be discussed and a mea-sure for the thermodynamic performance is introduced. Then in Chapter 3 we will givethe governing equations and we will set the mathematical framework for our small-parameter asymptotics. Chapter 4 is concerned with the systematic development of thetheory of thermoacoustics in slowly-varying two-dimensional pores. We will show howthe leading-order, first-order, and second-order terms with respect to the acoustic Machnumber can be derived, which includes the derivation of the second harmonics and

the streaming terms. We will also show how under certain simplifying assumptionsanalytic expressions can be obtained. In Chapter 5 the results from Chapter 4 will begeneralized to arbitrary three-dimensional slowly-varying pores. Next in Chapter 6 thethermoacoustic equations are implemented and validated experimentally for severaltypes of standing-wave devices: a thermoacoustic couple, a thermoacoustic refrigera-tor, and a thermoacoustic prime mover. We will show how the performance is affectedby the operating conditions and parameters connected to geometry, stack material, andworking gas. Finally we will also show what kind of streaming velocity profiles canoccur. In Chapter 7 a specific type of traveling-wave prime mover is modeled and im-plemented numerically. An optimization routine is developed that for given systemparameters computes the ideal geometry. Moreover, we we will show how the perfor-mance is affected by variation of various parameters. Then in Chapter 8 an evolutionequation will be derived that predicts the development of shock waves near resonance,both in a closed tube and in interaction with a thermoacoustic stack. Lastly, in Chapter9 we give some conclusions, discuss our results, and give some suggestions for futurework.

1.6 Literature review

While Rotts theory of thermoacoustics [91,115121] was still limited to two-dimensionalpores or three-dimensional cylindrical pores, it was Arnott et al. [5] who first extendedRotts theory to arbitrary three-dimensional cross-sections, although nonlinear termswere not yet included. We have extended their results by allowing a slow variation in


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14 1.6 Literature review

the pore cross-section in axial direction. Furthermore, we also allow temperature de-pendence of speed of sound, specific heat, viscosity, and thermal conductivity, and wehave eliminated the restriction of steady pore-wall temperature. Additionally we havealso incorporated quadratic nonlinearities such as the second harmonics and streamingterms into the analysis.

There are numerous articles that have analyzed more specific choices for the stackpore geometry, shown in figure 1.12. The parallel-plate geometry has been investigatedmost extensively, in particular by Swift [131] and Rott [115], who also investigated cylin-drical cross-sections. Stinson [128] and Roh et al. [113] derived simultaneously expres-sions for rectangular cross-sections and Stinson and Champoux [129] used results ofHan [53] to solve the equations for triangular cross-sections. Lastly, we mention Swift

and Keolian [136] who calculated the thermoacoustic behavior for so-called pin-arraystacks, consisting of a hexagonal array of pins aligned in axial direction.

(a) Parallel plates (b) Circular (c) Rectangular (d) Triangular (e) Pin array

Figure 1.12: Possible stack pore cross-sections.

Previous treatments with variable cross-sections have always been restricted to widelyspaced pores [98, 117], whereas in this thesis no such assumption is made. Althoughvariable cross-sections occur mostly within the main resonator tube, they may also occurin the narrow stack pores. A more general formulation would allow stack geometriesother than collections of arbitrarily shaped pores. We mention Roh et al. [114], who in-troduce tortuosity and viscous and thermal dynamic shape factors to extend single-porethermoacoustics to bulk porous medium thermoacoustics. Furthermore, there exists avast amount of papers on flow through porous media with random or stochastic prop-erties, that could also be applied to thermoacoustic configurations. Auriault [13] givesa clear overview of various techniques that can be used:

Statistical modeling [69];

Self-consistent models [157];

Volume-averaging techniques [103]; Method of homogenization for periodic structures [12].

In addition Auriault [13] gives a short explanation of how the method of homogeniza-tion can be applied to analyze heat and mass transfer in composite materials. A detaileddiscussion of methods and results from the theory of homogenization and their appli-cations to flow and transport in porous media can be found in [58]. Another approachis demonstrated by Kaminski [63], who combines the homogenization approach witha stochastic description of the physical parameters to analyze viscous incompressibleflow with heat transfer.


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Introduction 15

In our analysis, we assume a steady-state situation in which the variables oscillateat (integer multiples of) the fundamental frequency (cf. [5, 119, 131]), which is not al-ways the case. Prime movers require a sufficiently high temperature difference before itgoes into self-oscillation. Atchley et al. have given a standing-wave analysis of this phe-nomenon and measured the complete evolution of the quality factor of a prime moverfrom below, through, and above onset of self-oscillation [69].

The use of a dimensionless framework is not very wide-spread. Olson and Swift [97]use dimensionless parameters to analyze thermoacoustic devices, but without trying toconstruct a complete theory of thermoacoustics. Instead dimensionless numbers areused to reduce the number of independent parameters in their experiments and forscaling purposes.

There have been many observations [14, 100, 132] demonstrating that at high am-plitudes measurements deviate significantly from predictions by linear theory. Stream-ing, turbulence, transition effects, higher harmonics, and shock waves are mentioned asmain causes for these deviations. Streaming [75,95] refers to a steady mass-flux densityor velocity, usually of second order, that is superimposed on the larger first-order oscil-lations. With the addition of a steady non-zero mean velocity, the gas moves throughthe tube in a repetitive 102 steps forward, 98 steps backward manner as describedby Swift [135]. Streaming is important as a nonzero mass flux can seriously affect theperformance of thermoacoustic devices. It can cause convective heat transfer, whichcan be a loss, but it can also be essential to transfer heat to and from the environment.Backhaus and Swift [14], in their analysis of a traveling-wave heat prime mover, showhow streaming can cause significant degradation of the efficiency.

The concept of mass streaming has been studied by many authors, (see e.g. [17, 50,

52, 98,117, 148]), but restricted to simple geometries such as straight two-dimensional orcylindrical pores, although Olson and Swift [98] do allow slowly varying cross-sectionsin the tube. Moreover, they show that variable cross-sections can occur in practicalgeometries and can be used to suppress streaming; a suitable asymmetry in the tubecan cause counter-streaming that balances the existing streaming in the tube. Baillietet al. [17], Rott [117], and Olson and Swift [98] also take into account the temperaturedependence of viscosity and thermal conductivity, although the latter two only considerwidely spaced pores.

Higher harmonics oscillate at integer multiples of the fundamental frequency, andcan become quite important at high amplitudes. Atchley and Bass [10], noticed exper-imentally that the generation of higher harmonics can cause highly nonlinear wave-forms that degrade the performance significantly. The impact of the harmonics is great-

est when excited near resonance, but Atchley and Gaitan [43] analyzed that this canbe suppressed by careful tuning of the resonator. There is no literature that systemat-ically includes the higher harmonics into the analysis, which Swift [135] describes asa formidable challenge and Rott [118] as a hopeless undertaking. Although a com-plete extension would require going up to fourth order in the asymptotic expansion,we have shown here that the second-harmonic pressure and velocity oscillations can beexpressed in terms of the first-harmonic oscillations.

It has been noticed that higher harmonics can interact together to form shock waves[59, 60]. Neglecting the nonlinear sound field in the stack, Gusev [49] analyzed shockformations using a nonlinear evolution equation. Modeling the stack by a reflectioncoefficient, we will derive a new evolution equation that predicts the development of


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16 1.6 Literature review

shock waves when the tube is excited near resonance.Turbulence arises at high Reynolds numbers, where the assumption of laminar flow

is no longer valid. It disrupts boundary layers and can negatively affect the heat trans-fer. Turbulence may also arise due to abrupt changes in the shape or direction of thechannels, which leads to the shedding of vortices [2, 89, 158] and can cause significantlosses. Swift [44, 135] gave some suggestions on how to include turbulence into themodeling, but they are by no means complete. It is therefore still a challenge to includesystematically the effects of turbulence into the analysis.

Finally we note that there is a long list of publications on the numerical simulation ofheat-driven and sound-driven thermoacoustic devices, focusing on both standing-waveand traveling-wave modes of operation. This list is too long to go into here, but it should

be mentioned that a large part of the thermoacoustic community uses the DeltaEC (orDeltaE) code [146, 147] which was developed at the Los Alamos National Laboratorybased on Swifts linear theory of thermoacoustics [135].


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Chapter 2

Thermodynamics

In this chapter we will explain the basic thermodynamics concepts that are at the basisof thermoacoustics. We will start by giving the fundamental laws of thermodynam-ics and show how they can be used to derive a thermodynamic efficiency and coefficientof performance that indicate how well a thermoacoustic prime mover or refrigerator per-form. Next we will shed some more light on the mechanisms behind the thermoacousticproduction of sound or heat by analyzing the thermodynamic cycle a gas parcel experi-ences.

2.1 Laws of thermodynamics

Thermodynamically speaking a thermoacoustic system is completely characterized bythe flows of heat and work, as shown in figure 2.1. Let TH be the temperature of a hotreservoir and TC the temperature of a cold reservoir. In a refrigerator acoustic work Wisused to generate a heat flow against the temperature gradient, removing heat QC at thelow temperature and releasing heat QH at the high temperature. In a prime mover workW is produced by transporting heat from the high to the low temperature, removingheat QH at the high temperature and releasing heat QC at the low temperature.

The energy flows within a thermoacoustic system are governed by the first and sec-ond law of thermodynamics. The first law concerns the conservation of energy and

describes the rate of change of the internal energy

U of a system with volume V, theheat and enthalpy flows into the system, and the work done by the system [35]:

U = Q + nHm pV+ P. (2.1)Here n represents the molar flow rate entering the system, Hm represents molar en-thalpy, Q represents heat power, and P represents other forms of power done on thesystem. The summation is used to account for all the different sources of sound andmass that are in contact with the system and plus and minus signs are used to indicateflows into or out of the system, respectively.

The second law of thermodynamics states that any process that occurs will tend toincrease the total entropy of the system. Mathematically this can be expressed by the


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18 2.1 Laws of thermodynamics

Refrigerator

TH

TC

QH

W

QC

(a) Refrigerator or heat pump

Prime mover

TH

TC

QH

W

QC

(b) Prime mover

Figure 2.1: The flows of work W and heat Q inside (a) a thermoacoustic refrigerator or heatpump and (b) a thermoacoustic prime mover. The arrows are used to indicate the exchange ofheat and work between the thermoacoustic system and the environment.

inequalitySi 0, (2.2)

where Si represents the entropy production in the system. The second law can also beformulated using an equality [35],

S = Q

T+ nSm + Si. (2.3)

It states that the rate of change of entropy S of a thermodynamic system is equal tothe sum of the entropy change due to heat flows Q with temperature T, due to massflows nSm, and due to the irreversible entropy production in the system Si. Again thesummation signs are used to allow for a multitude of sound or mass sources.

In the thermoacoustic systems considered there is no mass flow into or out of thesystem, the volume of the system is constant, and the only work performed on thesystem is the acoustic work W. As a result, we find that the laws of thermodynamicsreduce to

U = QC QH + W, for a refrigerator or heat pump,QC + QH W, for a prime mover, (2.4)and

s =

QCTC

QHTH

+ Si, for a refrigerator or heat pump,

QCTC

+QHTH

+ Si, for a prime mover.

(2.5)

In our analysis we consider a (time-averaged) steady-state situation, so that we can putU = 0 and S = 0. It follows that

QC QH + W = 0, (2.6)


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Thermodynamics 19

for all thermoacoustic devices. In addition, since the entropy generated by the systemSi 0, we find

QCTC

QHTH

, for a refrigerator or heat pump, (2.7)

QCTC

QHTH

, for a prime mover, (2.8)

where equality can only be reached in the ideal situation when there are no irreversibleprocesses.

2.2 Thermodynamic performance

2.2.1 Refrigerator or heat pump

The performance of a refrigerator or heat pump is measured by the so-called coefficientof performance COP. Since a refrigerator and a heat pump each have a different goal,cooling versus heating, the coefficients of performance are defined differently also.

When analyzing a refrigerator we are interested in maximizing the cooling powerQC extracted at temperature TC, while at the same time minimizing the net requiredacoustic power W. On the other hand, a heat pump aims at maximizing the heatingpower QH at temperature TH, while minimizing W. Therefore, the coefficients of per-formance COPre f (refrigerator) and COPhp (heat pump) are defined as the ratios of these

quantities,

COPre f :=QCW

, (2.9)

COPhp :=QHW

. (2.10)

The second law of thermodynamics limits the interchange of heat and work. Inparticular it follows from (2.6) and (2.7) that

COPre f =QC

QH QC TC

TH TC=: COPCre f, (2.11)

COPhp = QH

QH QC TH

TH TC=: COPChp . (2.12)

The quantity COPC is called the Carnot coefficient of performance , and gives the max-imal performance for all refrigerators or heat pumps. Using COPC we can introduce arelative coefficient of performance COPR as

COPR :=COP

COPC. (2.13)

Note that although COP and COPC may attain any nonnegative value, the relative co-efficient COPR will always be between 0 and 1.


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20 2.3 The thermodynamic cycle

2.2.2 Prime mover

The performance of a prime mover is measured by the so-called efficiency . A primemover uses heating power QH to produce as much acoustic power W as possible andthus the efficiency is defined as

=:W

QH. (2.14)

Using equation (2.6), we can write

=QH QC

QH. (2.15)

Applying equation (2.8), we obtain

TH TCTH

= 1 TCTH

=: C. (2.16)

By definition both the efficiency and the Carnot efficiency C are between zero andone. The same holds for the relative efficiency R which is defined as

R :=

C. (2.17)

Of course the efficiency is not the only criterion for a good thermoacoustic primemover. For certain applications one might be primarily interested in maximizing the

power output with the efficiency only of minor interest. Other competing criteria canbe cost, size, reliability, available materials, safety, and the complexity of the design.Naturally, the same holds for a refrigerator or heat pump.

2.3 The thermodynamic cycle

In Section 1.2 we gave an intuitive explanation for the basic mechanism of thermoa-coustics. In this section we will give a more detailed analysis, describing all the steps afluid parcel undergoes while it oscillates in a narrow pore. In particular we will discussthe time phasing of the thermodynamic effect and its relation to the pore size, basedon the analysis of Swift [131] and Ceperley [23]. Swift showed that for standing-wave

devices the fluid-parcel movements are very accurately described by the Brayton cy-cle [21]. Ceperley suggested to design thermoacoustic devices such that the thermody-namic cycle would match the ideal Stirling cycle [145] to reduce irreversible effects toa minimum. Liang [73] analyzed Ceperleys concepts further using a sinusoidal modelthat describes the thermodynamic cycles of gas parcels oscillating in a regenerator.

2.3.1 Standing-wave phasing

Suppose gas parcels oscillate with standing-wave phasing at a distance y from a solidplate supporting a temperature gradient. As shown in figure 2.2 the parcels will ex-perience a build-up of pressure (compression) and drop of pressure (expansion) while


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Thermodynamics 21

A B C D At

compressiondisplacement

expansiondisplacement

Figure 2.2: Velocity (-) and pressure ( ) as a function of time in a gas supporting astanding wave. The cycle consists of a compression/displacement step (A-B) and an expan-sion/displacement step (C-D). Depending on the distance between the gas and the solid heatingand cooling may occur as well.

undergoing displacement. Depending on the size of y relative the thermal penetrationdepth k, heat transfer may occur as well. We consider three cases:

No thermal contact (y k)When there is no thermal contact between the gas and the plate the gas parcelswill expand and compress adiabatically and reversibly and no heat transfer willtake place.

Perfect thermal contact (y k)In the first step the gas parcels are compressed and displaced towards the hot endof the plate and the parcel. At the same time the parcels will be heated if a largetemperature gradient is imposed (prime mover) and cooled if a small temperaturegradient is imposed (refrigerator). Because there is perfect thermal contact thecompression and heat exchange will take place simultaneously. In the next stepthe gas parcels are displaced back towards the cold end and the reverse effect takesplace. The parcels will be cooled for a prime mover and heated for a refrigerator.

Imperfect thermal contact (y k)Because of the distance between the gas and the plate, there is a time delay be-tween motion and heat transfer. As a result the gas parcels execute a four-stepcycle. In case of a refrigerator the parcels are compressed and displaced (A-B),cooled (B-C), expanded and displaced back (C-D), and heated (D-A). In case of aprime mover the heating and cooling steps are interchanged.

The acoustic power produced or absorbed by a gas parcel can be found from thearea

p dV in the pressure-volume diagrams. In figure 2.3 we see that heat can only be

converted in acoustic power or vice versa if there is a time delay between the compres-sion or expansion and the heat exchange. This thermodynamic cycle is what is knownas the idealized Brayton cycle. This is the reason why in standing-wave based devices


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volume

pressure

y k

volume

pressure

y k

volume

pressure

y k

A

B C

D

(a) prime mover

volume

pressure

y k

volume

pressure

y k

volume

pressure

y k

D

C B

A

(b) refrigerator

Figure 2.3: Schematic pressure-volume cycle executed by the gas parcel in (a) a standing-wave

prime mover and (b) a standing-wave refrigerator. Only when the gas parcels are about a thermalpenetration depth away from the plate, pressure and volume variations are out of phase and network is performed.

stacks are used with stack pores that have radii of several penetration depths. The poorthermal contact is crucial for the thermoacoustic effect to occur, but it also gives rise toirreversible heat transfer and friction, which has a negative effect on the efficiency. Wewill show below that in traveling-wave devices we can have perfect thermal contact andthus potentially higher efficiencies.

2.3.2 Traveling-wave phasingSuppose now the gas parcels oscillate with traveling-wave phasing at a distance y kfrom a solid plate supporting a temperature gradient. As shown in figure 2.4 the parcelswill undergo a cycle consisting of four steps:

A-B First the gas parcel is compressed at constant temperature.

B-C Then the parcel is displaced towards the hot end at constant volume and simulta-neously it is heated (prime mover) or cooled (refrigerator).

C-D Next the parcel is expanded at constant temperature.


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Thermodynamics 23

A B C D At

compression expansion

cooling/heating

displacement

heating/cooling

displacement

Figure 2.4: Velocity (-) and pressure ( ) as a function of time in a gas supporting a trav-eling wave in perfect contact with the solid. The gas undergoes a Stirling cycle of compression,cooling/heating, expansion, and heating/cooling.

D-A Finally, the parcel is displaced back towards the cold end at constant volume andsimultaneously it is cooled (prime mover) or heated (refrigerator).

Note that there is a continuous exchange of heat between the gas and the solid duringthe displacement steps.

The acoustic power produced or absorbed by the gas parcel can be found from thearea

p dV in the pressure-volume diagrams. In figure 2.5 we see that heat is con-

verted into acoustic power and vice versa, even though y k (cf. figure 2.3). Thisthermodynamic cycle is what is known as the idealized Stirling cycle.

volume

pre

ssure

y k

DC

BA

(a) refrigerator

volume

pre

ssure

y k

AB

CD

(b) prime mover

Figure 2.5: Schematic pressure-volume cycle executed by the gas parcel in (a) a traveling-waverefrigerator and (b) a traveling-wave prime mover at perfect thermal contact with the wall. Onlywhen the gas parcels are about a thermal penetration depth away from the plate, pressure andvolume variations are out of phase and net work is performed.


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This is the reason why in traveling-wave based devices regenerators can be usedthat have perfect thermal contact between gas and solid. As there is no irreversible heattransfer much higher efficiencies can be obtained. It should be mentioned that becauseof the very narrow pores, large viscous losses may occur, which is why traveling-wavedevices are usually designed such that in the regenerator the velocities are small.

In figure 2.6 we have summarized the (idealized) four-step cycles the fluid parcelsundergo as they oscillate along the walls of the stack pores of a standing-wave primemover or refrigerator and the regenerator pores of a traveling-wave prime mover orrefrigerator.

1 - Compression

2 - Cooling

Q1

W1

3 - Expansion

4 - Heating

Q2

W2

(a) Standing-wave re-frigerator

1 - Compression

2 - Heating

Q1

W1

3 - Expansion

4 - Cooling

Q2

W2

(b) Standing-wave txprime mover

1 - Compression

2 - Cooling

Q1

W1

3 - Expansion

4 - Heating

Q2

W2

(c) Traveling-wave re-frigerator

1 - Compression

2 - Heating

Q1

W1

3 - Expansion

4 - Cooling

Q2

W2

(d) Traveling-wave txprime mover

Figure 2.6: Typical fluid parcels, near a stack plate, executing the four steps (1-4) of the thermo-dynamic cycle in (a) a stack-based standing-wave refrigerator, (b) a stack-based standing-waveprime mover, (c) a regenerator-based traveling-wave refrigerator and (d) a regenerator-basedtraveling-wave prime mover.


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Thermodynamics 25

2.3.3 Bucket-brigade effect

Usually the displacement of one fluid parcel is small with respect to the length of theplate. Thus there will be an entire train of adjacent fluid parcels, each confined to a shortregion of length 2x and passing on heat as in a bucket brigade (figure 2.7). Althougha single parcel transports heat Q over a very small interval, Q is shuttled along theentire plate because there are many parcels in series.

QC QHTC TH

x

Q

W

Figure 2.7: Work and heat flows inside a thermoacoustic refrigerator. The oscillating fluidparcels work as a bucket brigade, shuttling heat along the stack plate from one parcel of gas tothe next. As a result heat is transported from the left to the right, requiring acoustic power W.Inside a prime mover the arrows will be reversed, i.e. heat is transported from the right to the left

and acoustic power W is produced.


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Chapter 3

Modeling

3.1 Geometry

As mentioned in the introduction we consider devices of the form shown in figure3.1. The porous medium is modeled as a collection of narrow arbitrarily shaped poresaligned in the direction of sound propagation as shown in Fig. 3.2. Furthermore, weallow the pore boundary to vary slowly (with respect to the channel radius) in the di-rection of sound propagation, as shown in figure 3.2(b). For the analysis here we focuson what happens inside one narrow channel, which can represent a stack pore, but alsothe main resonator tube.

(a) straight duct (b) looped duct

Figure 3.1: Schematic view of two possible duct configurations: straight or looped.

In our analysis of the thermoacoustic effect we restrict ourselves to a pore and itsneighboring solid, where the system (x, r, ) forms a cylindrical coordinate system. LetAg(x) denote the gas cross-section and As(x) the half-solid cross-section at position x.Let g denote the gas-solid interface and t the outer boundary of the half-solid. Wechoose t and As such that there is no heat flux through t (for periodic structures t isthe centerline of the solid). The remaining stack pores and solid can then be modeled


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28 3.2 Governing equations

by periodicity. On g we write

Sg(x, r, ) := r Rg(x, ) = 0,

with Rg(x, ) the distance ofg to the pore centerline at position x and angle . Similarlyon t we write

St(x, r, ) := r Rt(x, ) = 0,where Rt(x, ) := Rg(x, ) + Rs(x, ) denotes the distance oft to the origin at positionx and angle .

(a) transverse cut of stack

solid

gasrx

Rg(x, )

Rs(x,)g

t

(b) longitudinal cut of one stack pore

Figure 3.2: Porous medium modelled as a collection of arbitrarily shaped pores.

3.2 Governing equations

The general equations describing the thermodynamic behavior in the gas are the well-known conservation laws of mass, momentum and energy [85]. Introducing the con-vective derivative

df

dt=

f

t+ v f,

these equations can be written in the following form:

mass :d

dt= v, (3.1)

momentum : dv

dt= p + + b, (3.2)

energy: d

dt= q p v + :v. (3.3)

Here is the density, v the velocity vector, p the pressure, b the body forces (gravity),q the heat flux, the specific internal energy, and the viscous stress tensor. For q and


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Modeling 29

we have the following relations

Fouriers heat flux model: q = KT, (3.4)Newtons viscous stress tensor: = 2D + ( v)I, (3.5)

where T the absolute temperature, K is the thermal conductivity, D = 12 (v + (v)T)the strain rate tensor, and and the dynamic (shear) and second viscosity.

In general the viscosity and thermal conductivity will depend on temperature. Forexample, Sutherlands formula [74] can be used to model the dynamic viscosity as afunction of the temperature

= re f 1 + Su/Tre f

1 + Su/T T

Tre f1/2 ,

where re f is the value of the dynamic viscosity at reference temperature Tre f and Suis Sutherlands constant. According to [26] the variation of thermal conductivity is ap-proximately the same as the variation of the viscosity coefficient. The thermodynamicparameters Cp, Cv, , and c may also depend on temperature (see Appendix A). Thetemperature dependence of and Kis particularly important, as it allows investigationof Rayleigh streaming: forced convection as a result of viscous and thermal boundary-layer phenomena.

As shown in Appendix B.2 (eq. (B.9)), the equations above can be combined and

rewritten to yield the following equation1 for the temperature in the fluid

CpTt

+ v T = T pt

+ v p+ (KT) + :v, (3.6)where Cp is the isobaric specific heat and the volumetric expansion coefficient as de-fined in Appendix A.

In the solid we only need an equation for the temperature. We have that the temper-ature Ts in the solid satisfies the diffusion equation

sCsTst

= (KsTs) , (3.7)

where Ks, Cs and s are the thermal conductivity, the isobaric specific heat and the den-sity of the solid, respectively.

Another useful equation, that can replace (3.3), describes the conservation of energyin the following way:

t

1

2|v|2 +

=

v

1

2|v|2 + h

KT v

+ v b, (3.8)

where h is the specific enthalpy. The details of the derivation leading to this equationcan be found in Appendix B.1.

The equations (3.1), (3.2), (3.6), and (3.7) can be used to determine v, p, T, and Ts.

1The double-dot product results in a scalar: A:B = i,j Ai jBji.


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30 3.2 Governing equations

Additionally one should add an equation of state that relates the density to the pressureand temperature in the gas. For the analysis here it is enough to express the thermody-namic variables , s, and h in terms of p and T using the thermodynamic equations(A.7)-(A.10) given in Appendix A. For the numerical examples given in Chapters 6 and7 it is necessary to make a choice and we choose to impose the ideal gas law

p = RspecT,

where Rspec = Cp Cv is the specific gas constant. However, the analysis presented hereis also valid for non-ideal gases and nowhere will we use this relation.

To distinguish between variations along and perpendicular to sound propagationwe will use a in the index to denote the transverse vector components. For example

the transverse gradient and transverse Laplace operator 2 are defined by

= ex

x+ , 2 =

2

x2+ 2 .

The equations will be linearized and simplified using the following assumptions:

the temperature variations along the stack are much smaller than the typical ab-solute temperature;

time-dependent variables oscillate with fundamental frequency .

At the gas-solid interface we impose the no-slip condition

v = 0 ifSg = 0. (3.9)The temperatures in the solid and in the gas are coupled at the gas-solid interface g bycontinuity of temperature and heat fluxes:

T = Ts ifSg = 0, (3.10a)K

T

n = Ks

Ts

n ifS

g = 0, (3.10b)

where n is a vector outward normal to the surface. The boundary t, with outwardsurface normal n, is chosen such that no heat goes through, i.e.

Ts n = 0 ifSt = 0. (3.11)Note that since Sg and St are vectors normal to the surfaces we can also replace thetwo flux conditions by

KT Sg = KsTs Sg ifSg = 0, (3.12)Ts St = 0 ifSt = 0. (3.13)


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Modeling 31

3.3 Scaling

To make the equations above dimensionless we apply a straightforward scaling proce-dure. First we scale the spatial coordinate by a typical pore radius Rg:

x = Rgx.

Note that because we consider narrow stack pores, the typical length scale L of radiusvariation is much larger than a radius. We therefore introduce the small parameter asthe ratio Rg/L between a typical radius and this length scale. At g we write

Sg(X, r, ) = r Rg(X, ) = 0, Rg(X, ) = RgRg(x, ), X = x = xL .

Similarly at t we have

St(X, r, ) = r Rt(X, ) = 0, Rt(X, ) = RgRt(x, ).Our formal assumption of slow variation has now been made explicit in the slow vari-

able X. As a typical time scale we use L/cre f, and we scale

t =L

cre ft.

Secondly, we rescale the remaining variables as well using characteristic values

u = cre fu, v = cre fv, p =

re fg (c

re f)2 p, =

re fcre f

Rg, (3.14a)

= re fg , T =(cre f)2

Cre fpT, s =

re fs s , Ts =

(cre f)2

Cre fpTs, (3.14b)

h = (cre f)2h, = (cre f)2, s = Cre fp s, b = gb, (3.14c)

=Cre fp

(cre f)2, Cp = C

re fp Cp, Cs = C

re fs Cs, c = c

re fc, (3.14d)

K= Kre f

K, = re f

, = re f

. (3.14e)

Thirdly, we observe that the system contains 19 parameters, in which 6 physical dimen-sions are involved. The Buckingham theorem [20] implies that the 19 parameters canbe combined into 13 independent dimensionless parameters. In Table 3.1 a possiblechoice is presented. Here the parameters , , k and s are the viscous penetrationdepths based on dynamic and second viscosity and the thermal penetration depths forthe fluid and solid, respectively.

=

2re f

re fg

, =

2re f

re fg

, k =

2Kre fg

re fg C

re fp

, s =

2Kre fs

re fs C

re fs

.


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34 3.4 Small-amplitude and long-pore approximation

Equations (3.16) and (3.17) show that the gravitational term b occurs as a second-ordereffect in the Mach number Ma. The equations (3.6) and (3.7) for the fluid and solidtemperature transform into

Cp

T

t+ u

T

X+ v T

= T

p

t+ u

p

X+ v p

+

2N2L (KT) + , (3.20)

sCsTst

=

2

2N2s (KsTs) . (3.21)

Furthermore the conservation of energy (3.8) transforms into

t

1

2|v|2 +

=

v

1

2|v|2 + h

2N2LT

W2ov

+

M2aFr

v b.

(3.22)Finally we have the following boundary conditions at the gas-solid interface. First

for the velocity we have

v = 0 ifSg = 0. (3.23a)Secondly, for the temperature we can write

T = Ts ifSg = 0, (3.23b)KT Sg = KsTs Sg ifSg = 0, (3.23c)Ts St = 0 ifSt = 0. (3.23d)

3.4 Small-amplitude and long-pore approximation

We assume a small maximal amplitude A of acoustic oscillations, i.e. the velocity, pres-sure, density and temperature fluctuations are small relative to their mean value, whichcan be used to linearize the equations given above. Since all variables are dimension-less, we can use the same A to linearize all variables, e.g. the relative pressure or velocityamplitude. Let f be any of the fluid variables (p, v, T, etc.) with stationary equilibriumprofile f0. Expanding f in powers of A using a Fourier series we write

f(x, t) = f0(x) + ARe

f1(x)eit

+ A2f2,0 (x) + A2Re

f2,2 (x)e

2it

+ , A 1.

Furthermore, for the velocity we assume v0 = 0. As the velocity was scaled with thespeed of sound, the obvious choice for A then becomes the acoustic Mach number Ma,


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Modeling 35

and we will therefore use the following asymptotic expansion:

f(x, t) = f0(x) + MaRe

f1(x)eit

+ M2a f2,0 (x) + M2a Re

f2,2 (x)e

2it

+ ,Ma 1. (3.24)

The first index is used to indicate the corresponding power of Ma, and the index behindthe comma is used to indicate the frequency of the oscillation (cf. [135]). The fluid vari-ables are built up from steady terms and harmonic modes that are integer multiples ofthe fundamental frequency. We assume that only the first harmonic f1 is excited exter-nally. It then follows from equation (3.26) below that the first-order oscillations excitethe second-order fluid variables by the combined effect of the second harmonic f2,2 , thatoscillates at twice the fundamental frequency, and the steady streaming part f2,0 . In thenext chapters we will show how both the streaming variables and the second harmonicscan be expressed in terms of the first harmonics and the equilibrium values.

Note that for the subsequent analysis we will only use the leading-order approxima-tion of the thermodynamic parameters c, , Cp, and Cv. Their indices will therefore beomitted.

Additionally, since we also assume 1, we may expand the perturbation variablesfi again to include powers of as well. However this would lead to messy derivations.Instead we will assume that Ma, so that the geometric and streaming effect can beincluded at the same order. In the end, when all the analysis has been performed, onecan still consider the limits Ma or Ma and neglect certain terms.

We will use an overbar to indicate time-averaging and brackets to indicate transverse

averaging in the gas or solid, i.e.

f(x) =

2

2/0

f(x, t) dt,

f(X, t) = 1Ag(X)

Ag(X)

f(x, t) dA, fs(X, t) =1

As(X)

As (X)

f(x, t) dA,

where Ag and As are the cross-sectional area at p

mathematical aspects of thermoacoustics.pdf

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